Induced Saturation in Graphs

• Induced saturation in graphs is defined by H-free graphs that acquire an induced copy of H upon any single edge modification, striking a critical balance.
• It employs combinatorial constructions and algebraic techniques, including trigraph frameworks and Cartesian products, to guarantee the fragile edge-critical property.
• Research on induced saturation examines existence, extremal functions, and applications in graph theory, offering deep insights into threshold phenomena.

Induced saturation in graphs generalizes the classical notion of graph saturation to the regime of induced substructures. A graph $G$ is $H$-induced-saturated if $G$ is $H$-free, but any addition or deletion of a single edge in $G$ creates an induced copy of $H$. This property highlights a threshold phenomenon: $G$ is precisely at the brink of containing $H$ as an induced subgraph—neither too sparse nor too dense, but critically balanced between forbidden and forced substructures. The paper of induced saturation encompasses both combinatorial constructions (finite and infinite), extremal functions (such as the induced saturation number), and the general existence landscape for such graphs.

1. Formal Definitions and Core Notions

Given fixed graphs $H$ ("target") and $G$ ("host"), $G$ is $H$-induced-saturated if:

1. $G$ contains no \emph{induced} subgraph isomorphic to $H$.
2. For every edge $e \in E(G)$, the graph $G-e$ contains an induced copy of $H$.
3. For every non-edge $f \in E(\overline{G})$, the graph $G+f$ contains an induced copy of $H$.

This can equivalently be phrased via trigraphs: a trigraph $T=(V,E_B,E_W,E_G)$, with black (forced edges), white (forced non-edges), and gray (optional/potential) edges is $H$-induced-saturated if no realization of $T$ contains induced $H$, but turning any forced edge into gray makes $H$ appear in some realization (Martin et al., 2011).

The induced saturation number, $\mathrm{indsat}(n,H)$, is the minimal number of gray edges in an $H$-induced-saturated $n$-vertex trigraph. When $\mathrm{indsat}(n,H)=0$, an $n$-vertex $H$-induced-saturated (simple) graph exists (Behrens et al., 2015).

2. Existence and Nonexistence: Families and Constructions

The existence of $H$-induced-saturated graphs is highly dependent on the choice of $H$. While classical (non-induced) $H$-saturated graphs exist for all $n \geq |V(H)|$, for induced-saturation, this is not universally true.

Positive existence results:

• Stars $K_{1,k+1}$, for $k\geq 2$ and sufficiently large $n$, have induced-saturated constructions as Cartesian products of cliques (Behrens et al., 2015, Axenovich et al., 2018).
• Matchings $kK_2$ yield induced-saturated graphs via multipartite blow-ups for large enough $n$ (Behrens et al., 2015).
• All odd cycles $C_{2k+1}$ of length at least 5 admit constructions, notably as the line graphs $L(K_{k,k})$ (Axenovich et al., 2018).
• For even cycles $C_{2\ell}$, sporadic constructions exist for $C_4$ (icosahedron), $C_6$ ($C_5\square C_5$), $C_8$ (dodecahedron), $C_{10}$ (computer-generated), but general constructions are missing for $\ell\geq 6$ (Fan et al., 30 May 2025).
• For paths $P_n$: No $P_4$-induced-saturated graph exists (Martin et al., 2011), but $P_2, P_3$ trivial cases exist; $P_6$ has an algebraic construction on 16 vertices (Raty, 2019); for all $n\geq 6$, explicit constructions (Dvořák’s twisted cycles) are given (Dvořák, 2020). Graphs for all $P_{3n}$ (for all $n$) are constructed combinatorially (Cho et al., 2019). For $n=5$, the question was open and resolved only recently.

Nonexistence:

• No $P_4$-induced-saturated graph exists (Martin et al., 2011, Behrens et al., 2015).
• For cliques and stable sets, induced-saturated graphs do not exist, since edge-flipping cannot create larger induced cliques or independent sets (Bonamy et al., 10 Jun 2025).

Infinite induced-saturated graphs exist for all finite $H$ that are neither cliques nor independent sets. Moreover, such graphs can be made "strongly induced-saturated" under all locally finite edge modifications (Bonamy et al., 10 Jun 2025).

3. Explicit Constructions: Methods and Techniques

Induced-saturated graphs are typically constructed using highly symmetric combinatorial or algebraic methods to guarantee both the $H$-freeness and fragile structure:

• Algebraic constructions: The $P_6$-induced-saturated graph of Räty uses the field $\mathbb{F}_{16}$, with adjacencies defined via coset differences and difference sets to enforce the desired local properties (Raty, 2019).
• Combinatorial products: Cartesian products of cliques, grids, and (for cycles) wheels or twisted cycles provide host graphs for stars, paws, and other sparse targets (Behrens et al., 2015, Axenovich et al., 2018).
• Automorphism-rich graphs: Vertex- and edge-transitive graphs such as the Kneser graph $K(n,2)$ (which is $P_6$-induced-saturated) and the Petersen graph allow strong reduction by symmetry (Cho et al., 2019).
• Graph expansion and gluing: For even cycles, construction combines high-girth graphs, canonical "territory" expansions, and identification/gluing strategies to achieve edge-criticality (Fan et al., 30 May 2025).
• Blow-ups and gadget gluing: For infinite graphs, the method is to iteratively "fix" bad pairs by attaching suitable gadgets, tracking priorities, and ensuring all pairs achieve the induced-saturation property in the limit (Bonamy et al., 10 Jun 2025).

The methods differ by the target $H$; for small $H$, direct finite constructions are usually feasible; for general $H$, especially cycles of large length or more complex trees, the existence and construction challenge remains formidable.

4. Extremal Functions and Quantitative Parameters

The paper of induced saturation introduces two main extremal parameters:

• The induced saturation number $\mathrm{indsat}(n;H)$: the minimum number of gray edges in a trigraph on $n$ vertices that is $H$-induced-saturated (Martin et al., 2011).
• The induced-saturation* number $\mathrm{indsat}^*(n;H)$: the minimum number of edges in an $H$-induced-saturated graph on $n$ vertices, if such a graph exists (Behrens et al., 2015).

Table: Key extremal values for selected $H$ (from (Behrens et al., 2015, Martin et al., 2011)) | $H$ | $\mathrm{indsat}(n;H)$ | $\mathrm{indsat}^*(n;H)$ | |-----------------|-----------------------|-------------------------------| | $K_m$ | $\mathrm{sat}(n,K_m)$ | classical | | $K_m-e$ | $0$ | varies | | $P_3$ | $0$ | $0$ | | $P_4$ | $1$ | not always achieved by a graph| | $K_{1,3}+e$ | $0$ (for $n\ge7$) | formula (piecewise in $n$) | | $K_{1,3}$ | $0$ (large $n$) | $2n$ or $2n-2$ |

For $P_4$, a sharp result is $\mathrm{indsat}(n,P_4)=\lceil (n+1)/3 \rceil$, achieved in trigraphs but not by honest graphs (Martin et al., 2011).

In many cases, explicit constructions matching bounds are known only for special graph orders and families. The function $\mathrm{indsat}^*(n;H)$ can be non-monotone in $n$, e.g., for the paw $K_{1,3}+e$.

5. Generalizations, Partial Results, and Unsettled Cases

Several generalizations and partial results guide ongoing research:

• Even cycles: For every even $C_{2\ell}$ with $\ell\ge3$, there exists a finite $C_{2\ell}$-free graph such that removing any edge creates an induced $C_{2\ell}$, but the full induced-saturated property (including edge-addition) for arbitrary even cycles is unsettled except for small cases (Fan et al., 30 May 2025).
• Paths: Complete existence for $P_n$-induced-saturated graphs is now affirmed for all $n\ge6$ (Dvořák, 2020), with infinite families for $P_{3n}$ (Cho et al., 2019), but explicit, sparse constructions for arbitrary $n$ remain an area of further investigation.
• Families and forbidden sets: For families of graphs $F$, it can occur that each $H\in F$ has $\mathrm{indsat}(n;H)=0$ but the family as a whole has positive induced saturation number—threshold and split graphs are examples (Behrens et al., 2015).
• Hypergraphs and directed graphs: Extending induced saturation concepts to these settings is proposed as an open problem (Bonamy et al., 10 Jun 2025).

Outstanding open questions (see (Fan et al., 30 May 2025, Bonamy et al., 10 Jun 2025)):

• Characterize all finite $H$ for which a finite $H$-induced-saturated graph exists.
• Find polynomial-size constructions for induced-saturated graphs for all even cycles.
• Determine the induced saturation number for wider classes of $H$ and graph families.

6. Infinite Induced-Saturated Graphs and Strong Saturation

Bonamy et al. provided the first comprehensive classification for infinite induced-saturated graphs: for any $H$ that is neither a clique nor an independent set, there exists a countable $H$-free graph such that any locally finite edge disturbance produces an induced $H$ (Bonamy et al., 10 Jun 2025). The construction is recursive, using gadgets for each bad (unfixed) vertex pair, and is robust to infinite but locally finite modifications.

For $H$ equal to a complete or empty graph, such strong induced-saturated graphs do not exist, since toggling edges does not produce larger cliques or independent sets.

This framework opens new directions, including simultaneous saturation for families, and adaptations to other discrete structures.

7. Connections, Applications, and Theoretical Implications

Induced saturation bridges extremal graph theory, saturation, and hereditary properties. By working with edge-flips rather than just additions, it exposes a higher sensitivity of hereditary graph properties to local structural perturbations. The trigraph framework elegantly connects with Boolean formulas: the induced-saturation number corresponds to the minimum number of unassigned variables in a DNF-saturated partial assignment (Martin et al., 2011).

Applications arise in the paper of threshold and split graphs, extremal constructions for codes in Hamming spaces, and elsewhere saturation and hereditary properties intersect. The highly structured forms of induced-saturated graphs also provide testbeds for conjectures and techniques in symmetry, combinatorics, and probabilistic graph methods.

A complete classification of induced-saturated graphs for all $H$—especially beyond classical objects like stars, cycles, and paths—remains a prominent challenge in modern extremal graph theory.